Transcript (continued) A S

BUS172 (Section 9 and10)
Instructor: Nahid Farnaz (NHN)
North South University
BUS172:
Introduction to Statistics
Chapter 4 (6th edition)
Probability
Chap 4-1
Important Terms




Random Experiment – a process leading to an
uncertain outcome
Basic Outcome – a possible outcome of a
random experiment
Sample Space – the collection of all possible
outcomes of a random experiment
Event – any subset of basic outcomes from the
sample space
Chap 4-2
Important Terms
(continued)

Intersection of Events – If A and B are two
events in a sample space S, then the
intersection, A ∩ B, is the set of all outcomes in
S that belong to both A and B
S
A
AB
B
Chap 4-3
Important Terms
(continued)

A and B are Mutually Exclusive Events if they
have no basic outcomes in common

i.e., the set A ∩ B is empty
S
A
B
Chap 4-4
Important Terms
(continued)

Union of Events – If A and B are two events in a
sample space S, then the union, A U B, is the
set of all outcomes in S that belong to either
A or B
S
A
B
The entire shaded
area represents
AUB
Chap 4-5
Important Terms
(continued)

Events E1, E2, … Ek are Collectively Exhaustive
events if E1 U E2 U . . . U Ek = S


i.e., the events completely cover the sample space
The Complement of an event A is the set of all
basic outcomes in the sample space that do not
belong to A. The complement is denoted A
S
A
A
Chap 4-6
Examples
Let the Sample Space be the collection of all
possible outcomes of rolling one die:
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then
A = [2, 4, 6]
and
B = [4, 5, 6]
Chap 4-7
Examples
(continued)
S = [1, 2, 3, 4, 5, 6]
A = [2, 4, 6]
B = [4, 5, 6]
Complements:
A  [1, 3, 5]
B  [1, 2, 3]
Intersections:
A  B  [4, 6]
Unions:
A  B  [5]
A  B  [2, 4, 5, 6]
A  A  [1, 2, 3, 4, 5, 6]  S
Chap 4-8
Examples
(continued)
S = [1, 2, 3, 4, 5, 6]

B = [4, 5, 6]
Mutually exclusive:

A and B are not mutually exclusive


A = [2, 4, 6]
The outcomes 4 and 6 are common to both
Collectively exhaustive:

A and B are not collectively exhaustive

A U B does not contain 1 or 3
Chap 4-9
Probability

Probability – the chance that
an uncertain event will occur
(always between 0 and 1)
0 ≤ P(A) ≤ 1 For any event A
1
Certain
.5
0
Impossible
Chap 4-10
Probability Rules

The Complement rule:
P(A)  1 P(A)

i.e., P(A)  P(A)  1
The Addition rule:

The probability of the union of two events is
P(A  B)  P(A)  P(B)  P(A  B)
Chap 4-11
A Probability Table
Probabilities and joint probabilities for two events A
and B are summarized in this table:
B
B
A
P(A  B)
P(A  B )
P(A)
A
P(A  B)
P(A  B )
P(A)
P(B)
P(B )
P(S)  1.0
Chap 4-12
Addition Rule Example
Consider a standard deck of 52 cards, with four suits:
♥♣♦♠
Let event A = card is an Ace
Let event B = card is from a red suit
Chap 4-13
Addition Rule Example
(continued)
P(Red U Ace) = P(Red) + P(Ace) - P(Red ∩ Ace)
= 26/52 + 4/52 - 2/52 = 28/52
Type
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Don’t
count the
two red
aces twice!
Chap 4-14
Conditional Probability

A conditional probability is the probability of one
event, given that another event has occurred:
P(A  B)
P(A | B) 
P(B)
The conditional
probability of A given
that B has occurred
P(A  B)
P(B | A) 
P(A)
The conditional
probability of B given
that A has occurred
Chap 4-15
Example 4.15
Product Choice: Ketchup and Mustard

a)
b)
c)
d)
75% of customers use mustard, 80% use
ketchup, and 65% use both.
What is the probability that a customer will use
at least one of these?
What is the probability that ketchup user uses
mustard?
What is the probability that mustard user uses
ketchup?
Draw a Joint Probability Table for this problem
(SOLVED IN CLASS)
Chap 4-16
Conditional Probability Example


Of the cars on a used car lot, 70% have air
conditioning (AC) and 40% have a CD player
(CD). 20% of the cars have both.
What is the probability that a car has a CD
player, given that it has AC ?
i.e., we want to find P(CD | AC)
Chap 4-17
Conditional Probability Example
(continued)

Of the cars on a used car lot, 70% have air conditioning
(AC) and 40% have a CD player (CD).
20% of the cars have both.
CD
No CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
P(CD  AC) .2
P(CD | AC) 
  .2857
P(AC)
.7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-18
Conditional Probability Example
(continued)

Given AC, we only consider the top row (70% of the cars). Of these,
20% have a CD player. 20% of 70% is 28.57%.
CD
No CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
P(CD  AC) .2
P(CD | AC) 
  .2857
P(AC)
.7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-19
Multiplication Rule

Multiplication rule for two events A and B:
P(A  B)  P(A | B)P(B)

also
P(A  B)  P(B | A)P(A)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-20
Multiplication Rule Example
P(Red ∩ Ace) = P(Red| Ace)P(Ace)
 2  4  2
    
 4  52  52
number of cards that are red and ace 2


total number of cards
52
Type
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-21
Statistical Independence

Two events are statistically independent
if and only if:
P(A  B)  P(A)P(B)


Events A and B are independent when the probability of one
event is not affected by the other event
If A and B are independent, then
P(A | B)  P(A)
if P(B)>0
P(B | A)  P(B)
if P(A)>0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-22
Statistical Independence Example


Of the cars on a used car lot, 70% have air conditioning
(AC) and 40% have a CD player (CD).
20% of the cars have both.
CD
No CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
Are the events AC and CD statistically independent?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-23
Statistical Independence Example
(continued)
CD
No CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
P(AC ∩ CD) = 0.2
P(AC) = 0.7
P(AC)P(CD) = (0.7)(0.4) = 0.28
P(CD) = 0.4
P(AC ∩ CD) = 0.2 ≠ P(AC)P(CD) = 0.28
So the two events are not statistically independent
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-24
Marginal Probability Example
P(Ace)
 P(Ace  Red)  P(Ace  Black) 
Type
2
2
4


52 52 52
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 4-25